【作者】 杨丽娟；

【导师】 马如云；

【作者基本信息】 西北师范大学 ， 基础数学， 2022， 硕士

【摘要】 本学位论文研究了几类带狄拉克δ脉冲函数的线性以及非线性周期边值问题的解的存在性.主要工作如下:1.研究了带狄拉克δ脉冲函数的线性二阶周期边值问题解以及正解的存在唯一性,其中c=c(x),h=h(x)为[0,1]上的连续函数,ci,hj∈R,i=1,2,…,p,j=1,2,…,q,p,q ∈ N.狄拉克脉冲 δ=δ(x)作用在给定的点0<x1<x2<…<xp<1与0<y1<y2<…<yq<1上.据我们所知,该部分工作首次得到了带有狄拉克脉冲的二阶周期边值问题的唯一解以及唯一正解,且在ci,hj=0的情形下,推广了 Torres[J.Diff.Equ.,2003]的工作.2.研究了带狄拉克δ脉冲函数的非线性二阶周期边值问题正解的全局结构.其中 r 为正参数,c(x)∈C([0,1],R),ci∈R,a,%∈C([0,+∞),[0,+∞)),i=1,2,…,p,j=1,2,…q;p,q ∈ N.狄拉克脉冲 δ=δ(x)作用在点0<x1<x2<…<xp<1 与 0<y1<y2<…<yq<1 上,通过运用全局分歧理论获得了该问题正解的全局结构.该部分工作与Xu等人[Topol.Methods Nonlinear Anal.,2014]相比,不仅获得了该问题正解的存在性,且刻画了正解集的全局结构.3.研究了带狄拉克δ脉冲函数的二阶周期共振问题解的存在性,其中 ci,pj∈ R,i=1,2,…,p,j=1,2,…,q,p,q ∈ N.狄拉克 δ 脉冲 δ=δ(x)作用在 0<x1<x2<.…<xp<1 与 0<y1<y2<…<yq<1上,p:[0,1]→R连续,非线性项g,Ii:R→R连续且不一定有界.通过运用Leray-Schauder原理,获得了该问题解的存在性.该部分工作将Kannan和Ortega[J.Diff.Equ.,1985]的不带有脉冲项的结果发展到带脉冲的情形.对于带脉冲项的研究,该部分工作将Drabek和Langerova[J.Math.Anal.Appl.,2015]的非线性项从有界发展到无界.更多还原

【Abstract】 This academic dissertation studies the existence of solutions to periodic boundary value problems for some classes of linear and nonlinear ordinary differential equations with Dirac delta impulsive functions.The main results are as follows:1.We study the existence and uniqueness of the solution and positive solution for the linear second order periodic boundary value problem with Dirac delta impulsive functions of the form:where c=c(x),h=h(x)are continuous functions on[0,1],ci,hj∈R,i=1,2,…,p,j=1,2,…,q,p,q∈N.The Dirac delta impulses δ=δ(x)are applied at given points 0<X1<x2<…<xp<1 and 0<y1<y2<…<yq<1.As far as we know,this part of the work first get the unique solution and unique positive solution for the linear second order periodic boundary value problem with Dirac delta impulses.And for ci,hj=0,we promoted the work of Torres[J.Diff.Equ.,2003].2.We study the global structure of positive solutions for the nonlinear second order periodic boundary value problem with Dirac delta impulsive function of the form:where r>0 is a positive parameter.c(x)∈C([0,1],R),ci∈R,a,aj∈C([0,+∞),[0,+∞)),i=1,2,…,P,j= 1,2,…,q,p,q∈N.The Dirac delta impulses δ=δ(x)are applied at given points 0<x1<x2<…<Xp<1 and 0<y1<y2<…<yq<1.We obtain the global structure of positive solutions for this problem by using global bifurcation theory.Compared with Xu et al.in[Topol.Methods Nonlinear Anal.,2014],this part of the work not only obtain the existence of positive solutions for being studied problem,but portray the global structure of the positive solutions set.3.We study the existence of solutions for second-order periodic resonant problem with Dirac delta impulsive functions of the form:where ci,pj∈R,i=1,2,…,p,j=1,2,…q,p,∈N.The Dirac delta impulsesδ=δ(x)are applied at given points 0<x1<x2<…<xp<1 and 0<y1<y2<…<yq<1.p:[0.1]→R is continuous,nonlinearity g,Ii:R→R is continuous and may be not bounded.By using the Leray-Schauder principle,we obtain the existence of solutions for this problem.In this part,we development the results of without impulses in Kannan and Ortega[J.Diff.Equ.,1985]to the case of having the impulses.For the studying with impulses,our result develop the nonlinearity of Drabek and Langerova[J.Math.Anal.Appl.,2015]from bounded to unbounded.更多还原